There's a slight difference in Portuguese about taking and bringing things. For example of let's say a party.
If you are to take something to a party such as.. bread? You are 'bringing' it to the party.

I would like to add that, just like in English, it depends on whose perspective one is taking. If someone is already at the party, they see another guest bring bread to the party, yet if you decide to provide bread at the party when you go there, you will take bread before you rush out of your house to go to the party. Vou levar o pão se ainda não saí de casa, e o hospedeiro vai ver-me trazer o pão.

Informal English is inconsistent in the use of 'take' and 'bring', so it is confusing for English speakers. When I learned these words in Spanish and Portuguese, I was taught that 'trazer/bring' means 'to move toward the speaker', and 'levar/take' means 'to move away from the speaker'. So I can never say 'Eu trago' because I cannot move away from myself. Can a native speaker confirm this?

I am sorry, but that explanation of «trazer» and «levar» is incorrect. Think of it as in English, "to bring" is focusing on (movement towards) the destination, and "to take" is focusing on (movement away) from the starting point; it has nothing to do with speakers. I can say, «Trago pão do mercado todos os dias.» = "I take bread from the market everyday [the implication is that the bread was bought]." I can also say, «Sempre levo um bolo para as festas de aniversário.» = "I always bring a cake to the birthday parties." I hope this clears things up! :)

And I think it is somewhat a cognate to German 'tragen' meaning something like 'transportar, carregar' and even 'trajar' - a cognate again. Eu trago = ich trage, eu trouxe = ich trug... (an also irregular conjugation).